A key feature of this polar duality (see e.g. [1]}, [2]}) is that if \(\Sigma \subset {\mathbb {H}}^3\)  is a strictly convex surface in \({\mathbb {H}}^3\) , then the dual surface \(\Sigma ^*\subset \mathbb {DS}^3\)  is also strictly convex, with induced metric corresponding to the third fundamental form of \(\Sigma \) , and third fundamental form corresponding to the induced metric of \(\Sigma \) . As a consequence, Question \(W^{*thin}_{{\mathbb {H}}^3}\)  can be considered as a direct analog of Question \(W^{thin}_{{\mathbb {H}}^3}\)  but in the de Sitter space rather than in \({\mathbb {H}}^3\) . Moreover, the behavior of the duality on subsets with “thick” ideal boundary suggests that Question \(W^{*thick}_{{\mathbb {H}}^3}\)  can be considered as (close to) an analog of Question \(W^{imm}_{{\mathbb {H}}^3}\)  but in the de Sitter space, and conversely.
